Rationalize The Denominator(With Examples)

Sometimes during calculation, you come across fractions having irrational numbers as their numerators or denominators. As it’s a convention to have the denominator of a fraction as a rational number, therefore, in cases where the denominator of a fraction is an irrational number, you have to make sure that its denominator should have a rational number.

The procedure of converting the irrational denominator into a rational denominator is called rationalizing the denominator. The expression or a number that is used to convert an irrational denominator into a rational denominator is called a rationalizing factor.

What is Rationalizing?

In mathematics, rationalizing means converting an irrational number or an irrational expression into a rational number or a rational expression. The process of conversion from an irrational value to a rational value is done by multiplying an irrational number or an expression by another irrational number or an irrational expression. The irrational number or an irrational expression that is multiplied is called the rationalizing factor (RF).

For example, to rationalize $\sqrt{2}$, you need another $\sqrt{2}$ so that you get $\sqrt{2} \times \sqrt{2} = 2$.

Note: $\sqrt{2}$ is an irrational number and $2$ is a rational number.

Therefore, the rationalizing factor of $\sqrt{2}$ is $\sqrt{2}$.

Similarly, to rationalize $\sqrt[3]{3}$, you need another number $\sqrt[3]{9}$ so that you get $\sqrt[3]{3} \times \sqrt[3]{9} = \sqrt[3]{27} = 3$.

Here $\sqrt[3]{3}$ is an irrational number whereas $3$ is a rational number.

What Does Rationalize The Denominator Mean?

Rationalizing the denominator means the process of removing an irrational number from the denominator of a fraction. The convention is that the denominator of a fraction should always be a rational number. Further by doing so, we bring the fraction to its simplest form.

For example, $\frac {1}{\sqrt {2}}$ after rationalizing the denominator becomes $\frac {\sqrt{2}}{2}$.

Similarly, $\frac {1}{\sqrt[3]{2}}$ after rationaling the denominator becomes $\frac {\sqrt[3]{4}}{2}$.

Methods of Rationalizing The Denominator

There are two common ways of rationalizing the denominator of a fraction depending on the type of expression present in the fraction.

These two ways are

• Using the conjugate of an irrational number
• Using an algebraic identity

Rationalize The Denominator Using The Conjugate of Irrational Number

A conjugate of an irrational expression or a similar irrational number is a similar irrational expression with an opposite sign. If the sign with an irrational number is positive then the sign with the conjugate will be negative and vice-versa.

For example, the conjugate of $3 + \sqrt{2}$ is $3 – \sqrt{2}$ and the conjugate of $3 – \sqrt{2}$ is $3 + \sqrt{2}$.

Note: To write the conjugate of an irrational number, you change the sign of an irrational part not that of the rational part.

For example, conjugate of $2 + \sqrt{3}$ is $2 – \sqrt{3}$ and not the $-2 + \sqrt{3}$. Similarly, $-2 – \sqrt{3}$ is not the conjugate of $2 + \sqrt{3}$.

In the process of rationalizing a denominator, the conjugate is the rationalizing factor. The process of rationalizing the denominator with its conjugate is as follows.

Step 1: Multiply both the denominator and numerator by a suitable conjugate that will remove the irrational number or irrational expression in the denominator

Step 2: Make sure that all the irrational numbers in the given fraction are in their simplified form

Step 3: Simplify the fraction further, if required

Examples

Let’s consider some examples to understand the process of rationalizing the denominator using the conjugate of an irrational number.

Ex 1: Rationalize the denominator of the fraction $\frac {1}{3 + \sqrt{2}}$.

Conjugate of $3 + \sqrt{2}$ is $3 – \sqrt{2}$.

Therefore, the rationalizing factor is $3 – \sqrt{2}$.

Multiplying the numerator and the denominator of the fraction by the rationalizing factor (RF).

$\frac {1}{3 + \sqrt{2}} \times \frac {3 – \sqrt{2}}{3 – \sqrt{2}} = \frac {1 \times \left(3 – \sqrt{2}\right)}{\left(3 + \sqrt{2}\right)\left(3 – \sqrt{2}\right)}$

$ = \frac {3 – \sqrt{2}}{3^{2} – (\sqrt{2})^{2}}$

Note: ${\left(3 + \sqrt{2}\right)\left(3 – \sqrt{2}\right)}$ is of the form $\left(a + b\right)\left(a – b\right)$, where $a = 3$ and $b = \sqrt{2}$ and $\left(a + b\right)\left(a – b\right) = a^{2} – b^{2}$.

Therefore, $\frac {1 \times \left(3 – \sqrt{2}\right)}{\left(3 + \sqrt{2}\right)\left(3 – \sqrt{2}\right)} = \frac {3 – \sqrt{2}}{3^{2} – \left(\sqrt{2}\right)^{2}} = \frac {3 – \sqrt{2}}{9 – 2} = \frac {3 – \sqrt{2}}{7} = \frac {3}{7} – \frac {\sqrt{2}}{7}$.

Therefore, after rationalizing the denominator, $\frac {1}{3 + \sqrt{2}}$ becomes $\frac {3}{7} – \frac {\sqrt{2}}{7}$.

Ex 2: Rationalize the denominator of the fraction $\frac {2}{\sqrt{2} + 1}$

Conjugate of the denominator $\sqrt{2} + 1$ is $\sqrt{2} – 1$.

Multiplying the numerator and the denominator by $\sqrt{2} – 1$.

$\frac {2}{\sqrt{2} + 1} = \frac {2}{\sqrt{2} + 1} \times \frac {\sqrt{2} – 1}{\sqrt{2} – 1}$

$ = \frac {2 \times \left(\sqrt{2} – 1\right)}{\left(\sqrt{2} + 1\right) \times \left(\sqrt{2} – 1\right)}$

$ = \frac {2\sqrt {2} – 2 }{\left(\sqrt{2}\right)^{2} – 1^{2}}$

$ = \frac {2 \sqrt{2} – 2}{2 – 1} = \frac {2 \sqrt{2} – 2}{1} = 2 \sqrt{2} – 2$

Therefore, after rationalizing the denominator $\frac {2}{\sqrt{2} + 1}$ becomes $2 \sqrt{2} – 2$.

Rationalize The Denominator Using Algebraic Identities

Using algebraic identities is another way of rationalizing the denominator of a fraction. The algebraic identity used in the process is $a^{2} – b^{2} = \left(a – b\right)\left(a + b\right)$.

• For rationalizing $\left(\sqrt{a} – \sqrt{b} \right)$, the rationalizing factor is $\left(\sqrt{a} + \sqrt{b} \right)$
• For rationalizing $\left(\sqrt{a} + \sqrt{b} \right)$, the rationalizing factor is $\left(\sqrt{a} – \sqrt{b} \right)$

Examples

Let’s consider some examples to understand the process of rationalizing the denominator using algebraic identities.

Ex 1: Rationalize the denominator of the fraction $\frac {1}{\sqrt {3} – \sqrt{2}}$.

Rationalizing factor (RF) is $\sqrt {3} + \sqrt{2}$.

Multiplying the numerator and the denominator of the fraction by $\sqrt {3} + \sqrt{2}$.

$\frac {1}{\sqrt {3} – \sqrt{2}} = \frac {1}{\sqrt {3} – \sqrt{2}} \times \frac {\sqrt {3} + \sqrt{2}}{\sqrt {3} + \sqrt{2}}$

$ = \frac {1\times \left(\sqrt {3} + \sqrt{2} \right)}{\left(\sqrt {3} – \sqrt{2}\right)\left(\sqrt {3} + \sqrt{2}\right)}$

$ = \frac {\sqrt{3} + \sqrt{2}}{\left(\sqrt{3}\right)^{2} – \left(\sqrt{2}\right)^{2}}$

$= \frac {\sqrt{3} + \sqrt{2}}{3 – 2} = \frac {\sqrt{3} + \sqrt{2}}{1} = \sqrt{3} + \sqrt{2}$

Ex 2: Rationalize the denominator of the fraction $\frac {5}{\sqrt {11} + \sqrt{7}}$.

Rationalizing factor (RF) is $\sqrt {11} – \sqrt{7}$.

Multiplying the numerator and the denominator by $\sqrt {11} – \sqrt{7}$.

$\frac {5}{\sqrt {11} + \sqrt{7}} \times \frac {\sqrt {11} – \sqrt{7}}{\sqrt {11} – \sqrt{7}}$

$ = \frac {5 \times \left(\sqrt{11} – \sqrt{7}\right)}{\left(\sqrt{11} + \sqrt{7}\right)\left(\sqrt{11} – \sqrt{7}\right)}$

$ = \frac {5\sqrt{11} – 5\sqrt{7}}{11 – 7} = \frac {5\sqrt{11} – 5\sqrt{7}}{4} = \frac {5\sqrt{11}}{4} – \frac {5\sqrt{7}}{4}$

Rationalize The Denominator With $3$ or More Terms

The same procedure that was followed to rationalize the denominator with $2$ terms is used but with a little variation. 

Let’s consider a denominator that three terms: $a + b + c$. 

First of all, rationalize the denominator with two terms, i.e., $a + b$ by multiplying with its conjugate $a – b$.

Now, the term $a + b$ reduces to a single term and then take $c$ to repeat the same process.

Note: An expression with three terms $a + b + c$ can be written as $\left(a + b\right) + c$. Now, according to the identity $\left(a – b\right)\left(a + b\right) = a^{2} – b^{2}$, $\left(\left(a + b\right) – c\right)\left(\left(a + b\right) + c\right) = \left(a + b\right)^{2} – c^{2}$. Therefore, the conjugate of $a + b + c$ is $\left(\left(a + b\right) – c\right)$.

Examples

Let’s consider an example to understand the procedure.

Ex 1: Rationalize the denominator of the fraction $\frac {1}{1 + \sqrt{2} + \sqrt{3}}$.

$\frac {1}{1 + \sqrt{2} + \sqrt{3}}$ can be written as $\frac {1}{\left(1 + \sqrt{2}\right) + \sqrt{3}}$.

Therefore, the rationalizing factor will be $\left(1 + \sqrt{2}\right) – \sqrt{3}$.

Multiplying the numerator and the denominator with $\left(1 + \sqrt{2}\right) – \sqrt{3}$

$\frac {1}{\left(1 + \sqrt{2}\right) + \sqrt{3}} \times \frac {\left(1 + \sqrt{2}\right) – \sqrt{3}}{\left(1 + \sqrt{2}\right) – \sqrt{3}}$

$ = \frac {1 \times \left(\left(1 + \sqrt{2}\right) – \sqrt{3}\right)}{\left(1 + \sqrt{2}\right) + \sqrt{3} \times \left(\left(1 + \sqrt{2}\right) – \sqrt{3}\right)}$

$ = \frac {1 + \sqrt{2} – \sqrt{3}}{\left(1 + \sqrt{2}\right)^{2} – \left(\sqrt{3}\right)^{2}}$

$ = \frac {1 + \sqrt{2} – \sqrt{3}}{1 + 2\sqrt{2} + 2 – 3 }$

$ = \frac {1 + \sqrt{2} – \sqrt{3}}{2\sqrt{2}}$

The denominator still contains an irrational number $\sqrt{2}$, which can be removed by multiplying the numerator and the denominator by its rationalizing factor, i.e., $\sqrt{2}$.

$ \frac {1 + \sqrt{2} – \sqrt{3}}{2\sqrt{2}} \times \frac {\sqrt{2}}{\sqrt{2}}$

$= \frac {\sqrt{2} + 2 – \sqrt{6}}{4}$

Therefore, $\frac {1}{1 + \sqrt{2} + \sqrt{3}}$ after rationalizing the denominator becomes $\frac {\sqrt{2} + 2 – \sqrt{6}}{4}$.

Conclusion

Rationalizing the denominator is the process of making the irrational denominator of a fraction rational so that the fraction can be reduced to its simplest form.

Practice Questions

1. Write the rationalizing factor of
   • $\sqrt{5}$
   • $-\sqrt{7}$
   • $2 + \sqrt{5}$
   • $1 – \sqrt{7}$
   • $-3 + \sqrt{2}$
   • $-2 – \sqrt{2}$
2. Rationalize the denominator of the following fractions
   • $\frac {4}{\sqrt{11}}$
   • $-\frac {3}{\sqrt{3}}$
   • $\frac {1}{1 + \sqrt{2}}$
   • $\frac {3}{3 – \sqrt{2}}$
   • $\frac {9}{-5 + \sqrt{3}}$
   • $\frac {2}{-2 – \sqrt{5}}$
   • $\frac {1}{1 + \sqrt{3} + \sqrt{5}}$
   • $\frac {3}{2 – \sqrt{2} + \sqrt{3}}$
   • $\frac {2}{2 + \sqrt{7} – \sqrt{5}}$

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties
• What is an Integer – Definition & Properties

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